MR. J. W. L. GrLAISHER ON THE THEORY OE ELLIPTIC FUNCTIONS. 517 
(54), viz. by help of the theorems 
[ e \„r~^ e ’ (55) 
; (56) 
whence, operating on (54) with e~ n and making «=0, we have at once 
a/ 7 r f 3:2 (a^— fx ) 2 (:r-f ^,) 2 'j ^ C kr& 7r 2 27TX 16n 2 7r 2 4'7ZV£ > 1 
+ 4,2 +6 - 4^2 + & c .t=Jl+2e"^cos^+2«“^cos^+&c. k 
which is (53) if we take n=\. 
Of the two lemmas (55) and (56) the truth of the second is seen at once, for 
■> d2 / ,72 1 M \ 
- "«-“=(l-^+n2» 4 *4-&c.)<r- 
= (l— mV + j^ »V— &c.)<r" 
and (55) is easily established, since « being put =0 after the performance of the 
differentiations. 
e da2 
e -a “ cos xu du 
= fY~ 
cos xu du 
✓ 7T _fi 
/> 4n2 
2n ,e * 
But the investigation is not elementary ; and if we assume a knowledge of the integral 
r 
e~ a2 * 2 cos2 bxdx—~e a2 
we may as well apply it directly to prove (53) by Fourier’s theorem as explained in §7, 
or employ it as Schellbach has done (‘Die Lehre von den elliptischen Integralen &c.,’ 
1864, p. 30). It does not seem to be easy to establish (55) without the aid of an integral ; 
for, expanding in ascending powers of x , we have to show that when a= 0, 
a 9 , p \ ✓*■/, a 2 , a* p \ 
e a \a a 3 + « 5 &C ')~2ny- 4rc 2 +32w 4 &c ’j ^ 
and, taking the first term only, although we see at once that 
.^0 i _ n2 & r* 
da2-_~ e da? 1 
a Jo 
11 du- 
f e~ n2u2 du=~ 
2 n 
yet 
- n ? d2 1 1 1 .2.% 2 1.2.3.- 
e — -s — + 
— &c.. 
which is divergent, and cannot apparently by any simple method be so transformed that 
its value when a=0 may be evident, without the intervention of an integral. Thus the 
method depending upon (55), though more direct, is not so elementary as that described 
in the Philosophical Magazine. 
